## Fluctuation-dissipation theorem for non-equilibrium quantum systems

### Mohammad Mehboudi

^{1,2}

### , Anna Sanpera

^{1,3}

### , and Juan M.R. Parrondo

^{4}

1Departament de F´ısica, Universitat Aut`onoma de Barcelona - E08193 Bellaterra, Spain

2ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain

3ICREA, Psg. Llu´ıs Companys 23, 08001 Barcelona, Spain

4Departamento de F´ısica At´omica, Molecular y Nuclear and GISC, Universidad Complutense Madrid, 28040 Madrid, Spain May 24, 2018

**The fluctuation-dissipation theorem (FDT) is**
**a central result in statistical physics, both for**
**classical and quantum systems. It establishes a**
**relationship between the linear response of a sys-**
**tem under a time-dependent perturbation and**
**time correlations of certain observables in equi-**
**librium. Here we derive a generalization of the**
**theorem which can be applied to any Markov**
**quantum system and makes use of the symmetric**
**logarithmic derivative (SLD). There are several**
**important benefits from our approach.** **First,**
**such a formulation clarifies the relation between**
**classical and quantum versions of the equilib-**
**rium FDT. Second, and more important, it fa-**
**cilitates the extension of the FDT to arbitrary**
**quantum Markovian evolution, as given by quan-**
**tum maps. Third, it clarifies the connection be-**
**tween the FDT and quantum metrology in sys-**
**tems with a non-equilibrium steady state.**

The first version of the fluctuation-dissipation theo- rem (FDT) was derived by Callen and Welton [1] and subsequently generalized by Kubo [2,3] in the context of linear response theory. Since then, it has been a crucial tool to investigate physical properties, such as trans- port, energy absorption and susceptibilities, of systems close to thermal equilibrium [4–6]. More recently, it has been proved useful to assess the multipartite entangle- ment of complex quantum systems at thermal equilib- rium [7], and out of thermal equilibrium [8]. The use- fulness of the FDT in parameter estimation and other related metrology problems is the subject of multiple studies [9–12].

Despite the fact that the FDT is so widely used, the standard FDT applies only to small perturbations around thermal equilibrium states [2–5]. There has been an intense activity in the last years to general- ize the FDT to classical systems far from equilibrium [6,13–16] or to verify it experimentally [17], and more recently to quantum systems [18,19]. Two main strate- gies have been followed in this pursuit. The first one

looks for correction terms in the original equilibrium FDT [20,21], whereas the second keeps the very mathe- matical structure of the theorem by redefining the mag- nitude conjugated to an external parameter [22,23].

Here, we adopt the second strategy to prove a FDT
for generic quantum Markovian systems. The key point
in our derivation is the use of the symmetric logarithmic
derivative (SLD), Λ_{λ}*, of a density matrix ρ** _{λ}*depending

*on a real parameter λ, defined as:*

(Λ*λ**ρ**λ**+ ρ**λ*Λ*λ*) ≡ 2 *∂*

*∂λ*^{0}
* _{λ}*0

*=λ*

*ρ**λ*^{0}*.* (1)
The SLD is an observable with zero average, hΛ*λ*i* _{λ}* =
Tr[Λ

*λ*

*ρ*

*λ*] = 0, as can be easily proved by taking the trace of the above equation. It is intimately re- lated to the quantum Fisher information (QFI), F

*= TrΛ*

_{λ}^{2}

_{λ}*ρ*

*, which plays a prominent role in metrol- ogy, since the uncertainty of any unbiased observable*

_{λ}*A (i.e. with hAi*

_{λ}*= λ), satisfies the Cr´*amer-Rao bound

*Var(A)*

_{λ}*≥ 1/F*

*[24–30]. In this paper we show that the SLD provides a novel definition of an observable con- jugated to an external parameter, which is extremely useful to derive a completely general FDT for quantum Markov systems and to relate previous versions of the FDT for classical and quantum systems.*

_{λ}We start by applying the SLD to the simplest case
of a fluctuation-dissipation relation for the static sus-
ceptibility. Consider a quantum system whose density
*matrix ρ**λ**depends on an external parameter λ. Taking*
*ρ*0as a reference state, we are interested on the change
*of the expected value of a generic observable B under*
*a small change in λ. More precisely, for small λ, the*
*expected value of a generic observable B can be written*
as:

*hBi*_{λ}*≡ Tr[Bρ*_{λ}*] ' hBi*_{0}*+ χ*^{s}_{B}*λ,* (2)
where

*χ*^{s}_{B}*≡ ∂**λ*|_{λ=0}*hBi**λ**= Tr [B ∂**λ*|_{λ=0}*ρ**λ**] ,* (3)
*is the static susceptibility of observable B. Using the*
SLD, the derivation of a fluctuation-dissipation relation

## arXiv:1705.03968v4 [quant-ph] 23 May 2018

is straightforward:

*χ*^{s}* _{B}* =1

2*Tr [B(Λ*0*ρ*0*+ ρ*0Λ0)] = 1

2*hBΛ*0+ Λ0*Bi*0 (4)
which is the symmetrized correlation between observ-
*ables B and Λ*0, since hΛ0i0 = 0. In the Appendix A
we show that Eq. (4), when particularized to a ther-
*mal state ρ**λ**= e*^{−β(H}^{0}^{−λA)}*/Z(λ), with β = 1/(kT ) and*
*Z(λ) ≡ Tre*^{−β(H}^{0}* ^{−λA)}* and expressed in the eigenbasis

*of H*

_{0}, yields the standard fluctuation-dissipation rela- tion.

We now turn to the case of a generic Markov evolu-
tion given by the composition of completely positive and
*trace preserving (CPTP) maps. Let ξ*_{λ}*(ρ) be a CPTP*
*map that depends on a parameter λ. We assume that*
*each map ξ**λ**has an invariant state π**λ**, i.e., ξ**λ**(π**λ**) = π**λ*.
*We study a small time-dependent perturbation λ(t) af-*
*fecting the invariant state π*0. More precisely, we con-
*sider the evolution, in discrete time steps t = 1, 2, . . . ,*

*ρ(t) = ξ*_{λ(t)}*◦ ξ*_{λ(t−1)}*◦ · · · ◦ ξ*_{λ(1)}*(π*0*).* (5)
*The linear response of an observable B can be written*
as:

*hB(t)i = hBi*0+

*t*

X

*t*^{0}=1

*φ**B**(t − t*^{0}*)λ(t*^{0}) (6)
*where hBi*_{0}*= Tr[π*_{0}*B], hB(t)i ≡ Tr[ρ(t)B], and φ*_{B}*(t −*
*t*^{0}*) is the response function of the observable B under*
*the perturbation λ.*

One can extend the above definition to the case of
*maps acting for a short time ∆t.* In the continuous
*limit, ∆t → 0, the sum in (6) is replaced by the integral*

*hB(t)i = hBi*0+
Z *t*

0

*dt*^{0}*φ*_{B}*(t − t*^{0}*)λ(t*^{0}*).* (7)
*The generalized susceptibility is defined as the Fourier*
*transform of the response function (φ**B**(t) is assumed to*
*vanish for t < 0 due to causality):*

*χ*_{B}*(ω) =*
Z ∞

0

*dt φ*_{B}*(t)e*^{iωt}*.* (8)
The generalized susceptibility has interesting proper-
*ties such as the Kramers-Kronig relation between the*
*real and imaginary parts χ**B**(ω) = χ*^{0}_{B}*(ω) + iχ*^{00}_{B}*(ω).*

When the evolution is unitary under the Hamiltonian
*H*0*− λ(t)A, χ*^{00}_{B}*(ω) is called absorptive part of the sus-*
ceptibility [2–5], since the energy absorbed by the sys-
*tem due to the perturbation is proportional to χ*^{00}_{B}*(ω).*

*The static susceptibility χ*^{s}* _{B}* can be related to the re-
sponse function by considering a constant perturbation

*λ(t) = λ for t ≥ 0 [2–5]. In this case hB(t)i → hBi*

_{λ}*when t → ∞ and Eq. (7) implies that the static sus-*ceptibility is the integral of the response function or,

equivalently, the generalized susceptibility at zero fre-
*quency: χ*^{s}_{B}*= χ**B**(ω = 0).*

*To obtain the FDT we calculate ρ(t) up to linear*
*terms in λ. For that, we write ξ*_{λ}*= ξ*_{0} *+ λξ*_{1} *+ . . .*
*(notice that ξ*_{1} is not a CPTP map) and the invariant
*state as π*_{λ}*= π*_{0}*+ λπ*_{1}*+ . . . The invariance of π** _{λ}* under

*the map ξ*

*implies:*

_{λ}*ξ*1*(π*0*) + ξ*0*(π*1*) = π*1 (9)
*and the SLD of π**λ* *with respect to λ at λ = 0 obeys*

*2π*1= Λ0*π*_{0}*+ π*0Λ0*.* (10)
Expanding the evolution equation (5) up to linear
terms, we obtain

*ρ(t) = ξ*^{t}_{0}*(π*0) +

*t*

X

*t*^{0}=1

*λ(t*^{0}*) ξ*^{t−t}_{0} ^{0}*◦ ξ*1*◦ ξ*_{0}^{t}^{0}^{−1}*(π*0)

*= π*0+

*t*

X

*t*^{0}=1

*λ(t*^{0}*) ξ*^{t−t}_{0} ^{0}*◦ ξ*1*(π*0) (11)
*where we have used the invariance of π*0 *under ξ*0. The
*expected value of B is*

*hB(t)i = hBi*0+

*t*

X

*t*^{0}=0

Trh

*B ξ*_{0}^{t−t}^{0}*◦ ξ*1*(π*0)i

*λ(t*^{0}*).* (12)
Comparing (12) with (6), we immediately get

*φ*_{B}*(t) = TrB ξ*_{0}^{t}*◦ ξ*_{1}*(π*_{0}) . (13)
Using (9) and (10):

*φ**B**(t) = TrB ξ*^{t}_{0}*◦ (π*1*− ξ*0*(π*1))

= Tr*Bξ*_{0}^{t}*(π*_{1}*) − Bξ*^{t+1}_{0} *(π*_{1}) = −Tr [∆B(t)π1]

= −1

2*Tr [∆B(t)(Λ*_{0}*π*_{0}*+ π*_{0}Λ_{0})]

= −1

2*h∆B(t)Λ*0+ Λ0*∆B(t)i*0 (14)
*where ∆B(t) = B(t + 1) − B(t) and B(t) = ˜ξ*_{0}^{t}*(B) is the*
*evolution of the observable B in the generalized Heisen-*
berg picture for quantum maps [31–33]. Here ˜*ξ*0(·) is
the adjoint map (not necessarily trace preserving) with
respect to the scalar product between operators given
*by the trace, i.e., Tr[Aξ*0*(B)] = Tr[ ˜ξ*0*(A)B], for all pair*
*of operators A and B.*

The fluctuation-dissipation relation for the static case
(4) is recovered from (14) if the correlations between
*B(t) and Λ vanish in the limit t → ∞:*

*χ*^{s}* _{B}*=

∞

X

*t*^{0}=0

*φ**B**(t*^{0})

= −1 2 lim

*t→∞**(hB(t)Λ*0+ Λ0*B(t)i*0*+ hBΛ*0+ Λ0*Bi*0)

= 1

2*hBΛ*0+ Λ0*Bi*0*.* (15)

Finally, the continuous-time version of theorem (14) is
*φ*_{B}*(t) = −*1

2*d*_{t}*hB(t)Λ*0+ Λ_{0}*B(t)i*_{0}*.* (16)
Eqs. (14) and (16) are our main result. These results
are the quantum counterparts of the non-equilibrium
classical FDT derived by Agarwal in [13] and revived
recently in [22, 23]. Notice that, in the classical sce-
nario, the conjugate variable is defined as the derivative
of the logarithm of the steady state probability distribu-
tion. On the quantum scenario, the non-commutativity
of observables does not allow to simply replace the clas-
sical conjugate variable with the derivative of the loga-
rithm of the density matrix. Nevertheless, the choice of
the SLD observable solves this non-commutativity issue.

In the Appendix Bwe prove that the latter expression
(16), when particularized to Hamiltonian dynamics and
*equilibrium states of the form π*_{λ}*= e*^{−β(H}^{0}^{−λA)}*/Z(λ),*
is completely equivalent to the standard Kubo formula
[2–5]:

*φ**BA**(t) =* *i*

~Tr[A, π0*]B(t) =* *i*

~*h[B(t), A]i.* (17)
It is worth it to point out that, in the quantum case,
the Kubo formula does not yield a simple FDT; more
precisely, the response function cannot be expressed in
terms of the time derivative of a two-time correlation in
equilibrium. Such a relationship can only be derived in
the frequency domain for the absorptive part of the gen-
eralized susceptibility and the Fourier transform of the
correlation (see [3–5] and the AppendixB). On the con-
trary, our generalized FDT, namely Eqs. (14) and (16),
expresses the response function in terms of correlations
in the time domain and can be equally applied to classi-
cal and quantum systems. This uniform formulation is
possible due to the introduction of the SLD. In the clas-
sical case, the SLD coincides with the normal derivative
and consequently, for a thermal state with Hamiltonian
*H*0*− λA, the SLD is −β(A − hAi), whereas for a quan-*
*tum system with [H*0*, A] 6= 0, the SLD yields a nontriv-*
ial conjugated variable as shown below. We highlight
the usefulness of our FDT for quantum metrology ex-
plicitely through the examples that follow. However, let
us remark that such a link holds for any map that fits
our framework.

We illustrate the new generalized FDT with a simple example consisting of two harmonic oscillators with a modulated interaction. The Hamiltonian reads:

*H = H*_{0}*+ H**J*

*= ωa*^{†}_{1}*a*1*+ (ω + δ)a*^{†}_{2}*a*2*− J (t)(a*^{†}_{2}*a*1*+ a*^{†}_{1}*a*2*),*
(18)
*where a**i**and a*^{†}_{i}*are the ladder operators of i-th oscilla-*
*tor and δ denotes the frequency detuning between the*

*oscillators. We assume |J (t)| ω so that linear re-*
sponse theory holds at any time. First, we consider the
two harmonic oscillators placed in a bath at tempera-
*ture T , and examine the response of the system to a*
*perturbation J (t) = J*_{0} around the thermal equilibrium
*state ρ*_{0}*= exp(−βH*_{0}*)/Z. Classically, the susceptibility*
*is defined as ∂**J**hAi*_{J}*, with A = ∂**J**H = −(a*^{†}_{2}*a*1*+ a*^{†}_{1}*a*2)
being the conjugate variable. Notice that if the de-
*tuning is zero, then [H*0*, A] = 0, and the SLD reads*
Λ*J**(δ = 0) = −β(A − hAi** _{J}*) [34]. On the contrary, any

*δ 6= 0 forces that [H*0

*, A] 6= 0, and the SLD cannot be*anymore identified trivially as the conjugate variable

*A [12]. For a finite detuning, the SLD takes the form*Λ

*J*

*(δ) = C(δ)Λ*

*J*(0) (see [35,36]). This additional coef-

*ficient arises from the non-commutativity between H*

_{0}

*and A, and reads:*

*C(δ) ≡* *tanh(δ/2T )*

*δ/2T* *.* (19)

The use of the SLD allows us to show that the QFI is:

F_{J}*(δ) = C*^{2}*(δ) β*^{2}*Var(A)*_{J}*.* (20)
*The additional coefficient is bounded, 0 < C(δ) ≤ 1. It*
*has a maximum at δ = 0, and then it decreases mono-*
tonically with the detuning, therefore, the precision on
*the estimation of J decreases as the detune increases.*

We now address the problem of a time-dependent
*modulation of J (t) in a non-equilibrium environment*
induced by two thermal baths at different temperatures
*T*1*and T*2. A master equation that has been widely used
to describe the reduced state of the oscillators consists
of a Linblad equation [31,37–39]

˙

*ρ(t) = −i[H, ρ(t)] +* X

*i={1,2}*

D*i**[ρ(t)],* (21)

with two independent dissipators D* _{i}*
D

_{i}*[ρ(t)] = γ(N*

*+ 1)*

_{i}*a*_{i}*ρ(t)a*^{†}_{i}*− 1/2{a*^{†}_{i}*a*_{i}*, ρ(t)}*
*+ γN**i*

*a*^{†}_{i}*ρ(t)a**i**− 1/2{a**i**a*^{†}_{i}*, ρ(t)}*
*. (22)*
*Here N**i**= (exp[β**i**ω**i*] − 1)^{−1} is the mean occupation
*number of the i-th oscillator, and γ is the dissipation*
rate, which is assumed to be equal for both oscillators.

*The equation is only valid for small J (t)—compared to*
*ω, i.e., the system’s energy scale—and J (t) < δ [40].*

In particular, it does not predict the thermalization of
*the full two-oscillator system for finite J when T*_{1} =
*T*_{2}. *Obviously, for J (t) = 0, both oscillators evolve*
independently reaching a stationary state in which each
oscillator is at thermal equilibrium with its own bath:

*ρ*0= e^{−β}^{1}^{H}^{1}

Tr[e^{−β}^{1}^{H}^{1}] ⊗ e^{−β}^{2}^{H}^{2}

Tr[e^{−β}^{2}^{H}^{2}]*,* (23)

*where H**i* *denotes the free Hamiltonian of the i-th os-*
cillator. Since the model under study is quadratic in
creation and annihilation operators, one may equiva-
lently describe it by means of its covariance matrix
(CM) which contains only first and second moments
(i.e., Gaussian), notably simplifying the calculations.

The latter is described with the help of the quadratures
*x**i*= 1

√2*(a*^{†}_{i}*+ a**i**),* *p**i*= *i*

√2*(a*^{†}_{i}*− a**i*) (24)
which satisfy the standard commutation relation
*[x*_{j}*, p*_{k}*] = iδ** _{jk}*. In turn, the matrix elements of the

*CM are defined as follows: σ*

_{jk}*≡ hR*

_{j}*R*

_{k}*+ R*

_{k}*R*

_{j}*i /2 −*

*hR*

*j*

*i hR*

*k*

*i, with R*

*j*

*∈ {x*1

*, x*2

*, p*1

*, p*2}. The CM corre- sponding to the stationary state of the master equa- tion (21) is provided in the AppendixC, where we also find (i) the SLD, and (ii) the time evolution of all the quadratic observables. Note that according to Eq. (16), (i) and (ii) are the two key elements required for evalu-

*ation of φ*

*B*

*(t). Specifically, the SLD writes as:*

Λ0*= c*1*(x*1*x*2*+ p*1*p*2*) + c*2*(x*1*p*2*− p*1*x*2*),* (25)
*with c*_{1} *and c*_{2}being real numbers. Therefore, the SLD
is a non-local operator, hence the response of any local
observable vanishes.

*To proceed further, let us consider J (t) = J*_{0}(1 −
*cos ν t). It is convenient to define the time dependent*
counter part of the static susceptibility as follows:

*χ**B**(t) ≡ ∂**J*_{0}|_{J}

0=0*hB(t)i ,* (26)
which quantifies the deviations of the observable from
*its initial value, and does not depend on J*0. Note that
*for a constant perturbation (i.e., ν = 0) we recover*
*χ**B**(t → ∞) = χ*^{s}* _{B}*. In Fig.1

*(a) we depict χ*

*B*

*(t) versus*

*time for B = x*1

*x*2 evaluated at three different modula-

*tion frequencies ν = 0, ν = δ/2, and ν = δ. For the con-*

*stant perturbation, we observe that at short times χ*

*B*

*(t)*is oscillating but, as time passes, the system relaxes into

*a new steady state such that hB(t → ∞)i = hBi*

*. On*

_{J}*the contrary, for ν 6= 0, the system never relaxes to*a stationary state. In this case, the time dependent susceptibility can be approximated at sufficiently large times by:

*χ**B**(t) ≈ | χ**B**(ν) | cos(νt + α),* (27)
*where α = arctan Re χ**B**(ν)/Im χ**B**(ν). Effectively,*
*the generalized suceptibility |χ**B**(ν)|, maps onto the am-*
plitude of the oscillations of the time dependent sus-
*ceptibility, and its dependence on ν is illustrated in*
Fig.1(b). One can see that the dynamical susceptibility
*is a flat function for ν δ, increases sharply as ν turns*
out to be resonant with the detuning, and strongly de-
creases afterwards. A similar behaviour is observed for
*any other nonlocal observable B = {x*1*p*2*; p*1*x*2*; p*1*p*2}.

Futhermore, the metrological character of the FDT
when expressed through the SLD, can be seen by identi-
fying which is the best measurement to detect the small-
*est interaction coupling J (t) = J*_{0}. If we have no prior
*knowledge about J*_{0}, linear response theory ensures that
*T r[B(ρ(t) − ρ*_{0}*)] = J*_{0}*χ*_{B}*(t). Unavoidable, an expec-*
tation value bears a statistical error, *pVar(B)*_{0}*/n, by*
*repeating the measurement n times** ^{1}*. Thus, in order to
infer the value of the perturbation, the linear response
of the system must be larger than the statistical error,

*i.e., |J*0

*χ*

*B*

*(t)| >pVar(B)*0

*/n, setting a lower bound on*the smallest perturbation that one may detect by mea-

*suring B. The inverse of this lower bound defines the*

*sensitivity F (B)*

*t*

*of the observable B to the perturba-*

*tion J*0:

*F (B)**t**≡ |χ**B**(t)|/*p

*Var(B)*_{0}*/n.* (28)
Notice that the above definition is not restricted to the
model discussed here. It can be applied for estimation
of a generic parameter which is parametrized through a
divisible map as in (5). For a constant perturbation, as
*t → ∞, χ*_{B}*(t → ∞) ≈ Corr(B, Λ*_{0})_{0}. Substituting this
in Eq. (28), and using the Cauchy-Schwartz inequality,
*the upper bound on the sensitivity of B at t → ∞ is*
given by:

*F (B)*∞≤p

*nVar(Λ*0)0=p

*nF*0*,* (29)
with F0 *being the QFI evaluated at J*0 = 0. The
QFI has been previously used to quantify the ultimate
precision of parameter estimation in non-equilibrium
steady states of spin models [41]. The bound (29) is
saturated by performing a measurement in the Λ0 ba-
sis. In Fig. 2, we depict F (B)*t* for three observables
*B = {Λ*0*; x*1*x*2*; x*1*p*2}. Although no observable can
*overperform the sensitivity of the SLD at t → ∞, at*
shorter times this fails to be the case as shown in the
inset of Fig.2. More interestingly, the maximum value
*of F (B)*_{t}*is not necessarily achieved at t → ∞. A sim-*
ilar behavior is observed for a time-dependent pertur-
*bation J (t) = J*_{0}*(1 − cos νt).* Although in this case
inequality (29) does not apply, the FDT as expressed in
Eq. (16) links Λ_{0}with the response function. From the
oscillatory character of the time dependent susceptibil-
*ity, the optimal times to best estimate J*0 are given by
Eq. (27).

In summary, we have presented a novel formulation of the Fluctuation-Dissipation Theorem (FDT) in terms of the symmetric logarithmic derivative (SLD), which is completely general to describe the effect of a per- turbation on a quantum systems, within the linear re-

*1**In fact the statistical error at measurement time is Var(B)**t*,
but since we only keep the leading order, in the linear response
*regime we safely replace it by Var(B)*0.

0 100 200 300 400 500 600 - 50

0 50 100

0.1 0.5 1 5 10

0.01 0.10 1 10

*Figure 1: (color online) (a) Time dependent susceptibility of the observable B = x*1*x*2 versus time, and for three different
modulation frequencies. (b) The dynamical susceptibility as a function of the modulation frequency. See the text for details. In
*both panels we set ω = 1, δ = 0.1, T*1 *= 1, T*2*= 2, and γ = 0.01.*

*

*

*

*

*

*

*

*

*

**

*

**

*

***

******

*************************************

0 100 200 300 400 500 600

0 2 4 6 8 10

* * ** ****** *

0 5 10 15 20

*Figure 2: (color online) The sensitivity, F (B)**t* as a function
*of time for B = {x*1*x*2*; x*1*p*2; Λ0} for a constant perturbation
*J (t) = J*0 and otherwise same parameters as in Fig.1.

sponse. Such a formulation presents some clear ad- vantages. First, it unifies FDT for classical and quan- tum systems. Second, it can be straightforwardly ap- plied to any Markovian evolution by means of quantum maps. This permits the extension of the FDT to non- equilibrium dynamics. Third, it provides an explicit connection between the susceptibility of an observable to an external perturbation and the figure of merit in quantum metrology.

Our FDT can be used to generalize the detection of multipartite entanglement to the non-equilibrium steady states (NESS). Although at thermal equilibrium, or after quenching a thermal state, such relations are known [7, 8], there are less studies for a general NESS

state. In this regard, the sensitivity measure Eq. (28)—

which is a lower bound on the QFI [28, 42]—can be used to detect multipartite entanglement. Finally, our results prompt an interesting open question on the relationship between the quantum versions of FDT and fluctuation theorems (FT). Needless to mention that, the quantum versions of FT have been subject of extensive studies, that have produced a rich literature on the subject [43–45]. The relationship between FDT and FT has been made clear for classical systems.

On the other hand, the FT for quantum CPTP maps requires a condition that is not necessary in our derivation of the FDT, namely, the Kraus operators of the map must be ladder operators in a relevant basis:

the eigenbasis of the instantaneous stationary state
*π*0 for generic CPTP maps [46] or, for periodically
driven systems in contact with equilibrium reservoirs,
Floquet eigenstates [47] or displaced energy eigenstates
[48]. Our FDT suggests that one could obtain a more
general FT by using the SLD.

### Acknowledgments

We acknowledge financial support from the Span- ish MINECO projects FIS2013-40627-P, FIS2013- 46768, FIS2014-52486-R (AEI/FEDER EU), QIBEQI FIS2016-80773-P, Severo Ochoa SEV-2015-0522, and the Generalitat de Catalunya CIRIT (2014-SGR-966, 2014-SGR-874), and the Generalitat de Catalunya (CERCA Programme) and Fundaci´o Privada Cellex.

M.M. acknowledges financial support from E.U. under the project TherMiQ.

### A Quantum systems in thermal equilibrium

A relevant particularization of the static fluctuation dissipation relation Eq. (4) is the case of a quantum system with
*Hamiltonian H*_{0}*− λA at thermal equilibrium. In such a case the density matrix is ρ*_{λ}*= e*^{−β(H}^{0}^{−λA)}*/Z(λ), where*
*β = 1/(k*_{B}*T ) is the inverse temperature and Z(λ) ≡ Tre*^{−β(H}^{0}* ^{−λA)}* is the partition function of the system. The

*equilibrium static susceptibility of an arbitrary observable B under the perturbation λA is denoted as χ*

^{s}*and obeys Eq. (4). Furthermore, when the SLD is expressed in the eigenbasis of the unperturbed Hamiltonian H0*

_{B}*|ni = E*

*n*

*|ni,*one can obtain some interesting relationships for the equilibrium static susceptibility. It is convenient to rewrite the SLD using the Feynman’s formula:

*∂*

*∂λ*
_{λ=0}

*e*^{−β(H}^{0}^{−λA)}*= β*
Z 1

0

*ds e*^{−βH}^{0}^{(1−s)}*Ae*^{−βH}^{0}^{s}*.* (30)

*In the eigenbasis of H*0, the formula reads:

*hn|* *∂*

*∂λ*
_{λ=0}

*e*^{−β(H}^{0}^{−λA)}*|mi = β hn| A |mi*
Z 1

0

*ds e*^{−β[E}^{n}^{(1−s)+E}^{m}^{s]}*= hn| A |mie*^{−βE}^{m}*− e*^{−βE}^{n}*E**n**− E**m*

*,* (31)

*if E*_{n}*6= E**m**. Otherwise, the matrix element is β hn| A |mi e*^{−βE}* ^{n}*. Therefore:

*hn|* *∂*

*∂λ*
_{λ=0}

*ρ**λ**|mi = hn| A |mi* *p**m**− p**n*

*E**n**− E**m*

*,* *for E**n* *6= E**m**,*
*hn|* *∂*

*∂λ*
_{λ=0}

*ρ*_{λ}*|mi = p**n*

*β hn| A |mi − δ*_{mn}*Z*^{0}(0)
*Z(0)*

*,* *for E*_{n}*= E*_{m}*.* (32)

*Here p**n**= e*^{−βE}^{n}*/Z(0) is the population of level E**n* at equilibrium. Using Eq. (30):

*Z*^{0}(0)
*Z(0)* = *β*

*Z(0)*
Z 1

0

*ds Tr*h

*e*^{−βH}^{0}^{(1−s)}*Ae*^{−βH}^{0}* ^{s}*i

= *β*

*Z(0)*Tr*e*^{−βH}^{0}*A*

*= βhAi*_{0}*.* (33)

Eq. (1) can be written as:

*2 hn| A |mi* *p**m**− p**n*

*E*_{n}*− E**m*

*= (p**n**+ p**m**) hn| Λ*_{0}*|mi ,* (34)

*for E**n**6= E**m**. For E**n* *= E**m**by choosing the eigenbasis |ni such that A is diagonal in the eigen-subspaces of H*0we
have* ^{2}*:

*2β [hn| A |mi − hAi*0*] p**n**δ**mn**= 2 p**n**hn| Λ*0*|mi ,* (35)
Therefore:

*hn| Λ*0*|mi = 2hn| A |mi*
*p**n**+ p**m*

*p**m**− p**n*

*E**n**− E**m*

*for E**n**6= E**m**,*
*hn| Λ*_{0}*|mi = β [hn| A |mi − hAi*_{0}*] δ*_{mn}*for E*_{n}*= E*_{m}*.*
*Using this expression, the susceptibility of A reduces to:*

*χ*^{s}* _{A}*= 1
2

X

*n*

*p**n**hn| Λ*0*A + A Λ*0*|ni*

= 1 2

X

*n,m*

*p*_{n}*hn| Λ*_{0}*|mi hm| A |ni + hn| A |mi hm| Λ*_{0}*|ni*

= 1 2

X

*n,m*

*(p**n**+ p**m**) hn| Λ*0*|mi hm| A |ni*

*= β*

"

X

*n*

*p*_{n}*| hn| A |ni |*^{2}*− hAi*^{2}_{0}

#

+X

*n,m*

*p*_{m}*− p*_{n}*E**n**− E**m*

*| hn| A |mi |*^{2}*,* (36)

*2**This means that the eigenstates of H*0*are chosen such that for any two states |ni , |mi with the same energy, the criteria hn| A |mi 6= 0*
*holds only if m = n. Therefore, hn| A |mi = hn| A |mi δ**m,n*.

*where the last sum runs over all n and m with E**n* *6= E**m*. One can distinguish the Curie and van Vleck terms in
the FDT [4,5].

It is interesting to consider the observable ˜*A = Λ*_{0}*, i.e., the SLD of ρ*_{λ}*at λ = 0. Since [ ˜A, Λ*_{0}] = 0 and hΛ0i0= 0,
the susceptibility of ˜*A is equivalent to the QFI, i.e., we have χ*^{s}_{˜}

*A*= Var(Λ_{0})_{0}= F_{0}. Therefore, ˜*A identifies the most*
*sensitive observable to the perturbation (rather than A), and its sensitivity saturates the Cram´*er-Rao bound, that
is given by:

F0*= χ*^{s}_{A}_{˜}=X

*n*

*p*_{n}*hn| Λ*^{2}_{0}*|ni*

*= β*

"

X

*n*

*p*_{n}*| hn| A |ni |*^{2}*− hAi*^{2}_{0}

# + 2X

*n,m*

* p**m**− p**n*

*E*_{n}*− E**m*

^{2} *| hn| A |mi |*^{2}
*p*_{n}*+ p**m*

*.* (37)

### B Kubo relations

Our main results Eqs. (14) and (16) are FDTs for generic quantum Markov systems. One can recover the familiar
*Kubo quantum FDT for states π*_{λ}*= e*^{−β(H}^{0}^{−λA)}*/Z(λ) and Hamiltonian evolution. In this case, the Eq. (16) reads*
*(as before, we denote by φ**B**(t) the response function of observable B under the perturbation −λ(t)A):*

*φ**B**(t) = −*h ˙*B(t) Λ*0+ Λ0*B(t)i*˙ 0

2

= − *i*

2~*h[H*0*, B(t)] Λ*0+ Λ0*[H*0*, B(t)]i*0

= − *i*

2~Tr(Λ0*π*0*+ π*0 Λ0*) [H*0*, B(t)]*

= − *i*

2~Tr[Λ0 *π*_{0}*+ π*_{0}Λ_{0}*, H*_{0}*] B(t).* (38)
*To proceed further, we need an additional formula for the SLD. For any real function f (x):*

*[H*_{0}*− λA, f (H*_{0}*− λA)] = 0.* (39)

*Differentiating this equation with respect to λ and setting λ = 0, yields:*

*− [A, f (H*0)] +

*H*_{0}*,* *∂f (H*_{0}*− λA)*

*∂λ*
_{λ=0}

*= 0.* (40)

*Particularizing to f (x) = e** ^{−βx}* and using the definition of the SLD given by Eq. (1) the above equation reduces to:

*[A, π*_{0}] =1

2*[H*_{0}*, Λ*_{0} *π*_{0}*+ π*_{0}Λ_{0}*].* (41)

Introducing this last result into Eq. (38) we finally reach at:

*φ**B**(t) =* *i*

~Tr[A, π0*] B(t) =* *i*

~*h[B(t), A]i.* (42)

which is the standard Kubo formula. It is interesting to recall that the Kubo formula allows one to express the
*absorptive part of the susceptibility in terms of a two time correlation in the frequency domain. This is the so-called*
quantum FDT, which reads [4]:

*χ*^{00}_{B}*(ω) =* 1

~

tanh* β~ω*
2

*C*˜_{BA}*(ω),* (43)

where ˜*C**BA**(ω) is the Fourier transform of the symmetric correlation function C**BA**(t) = hB(t)A + AB(t)i*0*/2 −*
*hBi*0*hAi*0. The above equation does not have a simple correlate in the time domain for quantum systems. Only in
the classical case, ~ → 0, Eq. (43) reduces to:

*χ*^{00}_{B}*(ω) =* *βω*
2

*C*˜*BA**(ω),* (44)

which is equivalent to:

*φ*_{B}*(t) = −β* *d*

*dtC*_{BA}*(t).* (45)

Here we see the clear advantage of the introduction of the SLD conjugated variable Λ0: it allows us to express the
*response function in the time domain as a correlation, both in the quantum and in the classical case.*

### C The response function of coupled harmonic oscillators

Here with the help of the stationary state of the coupled harmonic oscillators, we find the response function of any
*observable B to the perturbation. The covariance matrix (CM) corresponding to the density matrix of Eq. (23),*
i.e., for a vanishing interaction, is described by the following diagonal matrix:

*σ*^{∞}* _{J =0}* =

*N*1+^{1}_{2} 0 0 0

0 *N*2+^{1}_{2} 0 0

0 0 *N*1+^{1}_{2} 0

0 0 0 *N*2+^{1}_{2}

*.* (46)

For a non-zero coupling, the CM is not diagonal anymore, as the coupling establishes correlations amongst the two harmonic oscillators. The corresponding CM is given by the following matrix [37],

*σ*^{∞}*= ζ*

*D + N*_{1}+^{1}_{2} *−δC* 0 *γC*

*−δC* *D + N*_{2}+^{1}_{2} *−γC* 0

0 *−γC* *D + N*_{1}+^{1}_{2} *−δC*

*γC* 0 *−δC* *D + N*_{2}+^{1}_{2}

*,* (47)

*with ζ =* * _{4J}*2

^{γ}*+γ*

^{2}

^{+δ}^{2}

^{2}

*+δ*

^{2}

*, D =*

^{2J}^{2}

^{(N}*2*

_{γ}^{1}

*+δ*

^{+N}^{2}

^{2}

^{+1)}

*, and C =*

^{J (N}*2*

_{γ}^{1}

*+δ*

^{−N}^{2}

^{2}

^{)}.

*Next, we need to identify (i) the SLD associated to J , and (ii) the time evolution of the desired observable B(t),*
under the unperturbed map. One can find (i) with the help of Eq. (47). We know that for such a Gaussian state,
Λ0 can be expressed as a linear combination of all the second order moments of the quadratures [35, 36], namely

Λ0*= d*1*x*^{2}_{1}*− (σ*^{∞}_{11})_{J =0}* + d*2*p*^{2}_{1}*− (σ*^{∞}_{33})_{J =0}* + d*3*[x*1*p*1*+ p*1*x*1]
*+ d*_{4}*x*^{2}_{2}*− (σ*_{22}^{∞})_{J =0}* + d*5*p*^{2}_{2}*− (σ*_{44}^{∞})_{J =0}* + d*6*[x*_{2}*p*_{2}*+ p*_{2}*x*_{2}]

*+ c*1*x*1*x*2*+ c*2 *x*1*p*2*+ c*3 *p*1*x*2*+ c*4 *p*1*p*2*,* (48)
*with d**j**s and c**j*s being coefficients which are to be determined. To this end, we make benefit of Eq. (4) of the main
*text, which states that for any observable B we have:*

*∂**J* |_{J =0}*hBi** _{J}*= 1

2hΛ0*B + BΛ*0i_{0}*.* (49)

Next, we imply this relation to all of the quadratic observables appearing in (48), i.e., we choose B ∈ {x^{2}_{1}*, x*^{2}_{2}*, x*1*p*1+
*p*1*x*1*, . . . }. With such choices of B, the left hand side of Eq. (49) can be evaluated by taking derivative from*
*the covariance matrix σ*^{∞}. Moreover, by using the Wick’s theorem for Gaussian distributions [49], one can easily
*simplify the right hand side and write it down in terms of the elements of σ*^{∞} as well. We shall start by focusing
*on the local terms. As an example, for B = x*^{2}_{1} we have:

1

2Λ0*x*^{2}_{1}*+ x*^{2}_{1}Λ0

0*= (∂**J**σ*_{11}^{∞})

*J =0* *,*

*⇒ 2d*1*(σ*^{∞}_{11})^{2}* _{J =0}*−

*d*

_{2}

2 *= 0,* (50)

*where we used the fact that σ*_{J =0}^{∞} , has no off-diagonal terms, as stated by Eq. (46). For B = p^{2}_{1}one finds a similar
*equation, with the change of coefficients d*1 *↔ d*2*. This implies that, d*1 *= d*2 = 0. In the same manner, one can
*find that d*_{4}*= d*5*= 0. For the other two local terms i.e., B ∈ {*^{1}_{2}*(x*1*p*_{1}*+ p*1*x*_{1}*),*^{1}_{2}*(x*2*p*_{2}*+ p*2*x*_{2})}, Eq. (49) leads to:

*d*3

*2(σ*_{11}^{∞})_{J =0}*(σ*_{33}^{∞})* _{J =0}*+1
2

*= 0,*
*d*6

*2(σ*_{22}^{∞})_{J =0}*(σ*_{44}^{∞})* _{J =0}*+1
2

*= 0,* (51)

*whence, d*3 *= d*6 = 0 as well. This confirms that the coefficients associated to local observables are zero, i.e.,
*d**i**= 0 ∀i. However, the non-local coefficients are non-zero. For B ∈ {x*1*x*2*, x*1*p*2*, p*1*x*2*, p*1*p*2}, using Eq. (49) yields:

*c*1*(σ*_{11}^{∞})_{J =0}*(σ*_{22}^{∞})_{J =0}*= (∂**J**σ*_{12}^{∞})_{J =0}*,* *c*2*(σ*_{11}^{∞})_{J =0}*(σ*_{44}^{∞})_{J =0}*= (∂**J**σ*^{∞}_{14})_{J =0}*,*

*c*_{3}*(σ*_{33}^{∞})_{J =0}*(σ*_{22}^{∞})_{J =0}*= (∂*_{J}*σ*_{32}^{∞})_{J =0}*,* *c*_{4}*(σ*_{33}^{∞})_{J =0}*(σ*_{44}^{∞})_{J =0}*= (∂*_{J}*σ*^{∞}_{34})_{J =0}*.* (52)
By using the symmetry in the covariance matrix one can simplify to arrive at:

*c*1*= c*4=* ∂*_{J}*σ*^{∞}_{12}
*σ*^{∞}_{11}*σ*^{∞}_{22}

*J =0*

*,* *c*2*= −c*3=* ∂*_{J}*σ*^{∞}_{14}
*σ*^{∞}_{11}*σ*_{44}^{∞}

*J =0*

*.* (53)

Replacing in Eq. (48) for the SLD yields:

Λ_{0}*= c*_{1}*(x*_{1}*x*_{2}*+ p*_{1}*p*_{2}*) + c*_{2}*(x*_{1}*p*_{2}*− p*1*x*_{2})

*= (c*_{1}*+ ic*_{2}*)a*^{†}_{1}*a*_{2}*+ h.c.* (54)

*The other key element (ii) is also easy to evaluate for any choice of B ∈ {x*^{2}_{1}*, x*^{2}_{2}*, x*_{1}*p*_{1}*+ p*_{1}*x*_{1}*, . . . }, because the time*
evolution shall be evaluated under the unperturbed map. To begin with, we remind that the time evolution of the
ladder operators are easy to find (see [31], for instance), and read as follow:

*a**j**(t) = e*^{(−iω}^{j}^{−}^{γ0}^{2}^{)t}*a**j**,* *a*^{†}_{j}*(t) = e*^{(iω}^{j}^{−}^{γ0}^{2}^{)t}*a*^{†}_{j}*,*

*a*^{†}_{j}*a*_{j}*(t) = e*^{−γ}^{0}^{t}*a*^{†}_{j}*a*_{j}*+ N (1 − e*^{−γ}^{0}^{t}*),* *a*_{j}*a*^{†}_{j}*(t) = e*^{−γ}^{0}^{t}*a*_{j}*a*^{†}_{j}*+ (N + 1)(1 − e*^{−γ}^{0}^{t}*).* (55)
*Therefore, by using the definition of x**j* *and p**j* quadratures, we find the first moments to evolve as:

*x*_{j}*(t) =* 1

√2*(a*^{†}_{j}*(t) + a*_{j}*(t)) = e*^{−γ/2t}*(cos(ω*_{j}*t)x*_{j}*+ sin(ω*_{j}*t)p*_{j}*) ,*
*p**j**(t) =* *i*

√2*(a*^{†}_{j}*(t) − a**j**(t)) = e*^{−γ/2t}*(− sin(ω**j**t)x**j**+ cos(ω**j**t)p**j**) .* (56)

In addition, the local second moments are:

*x*^{2}_{j}*(t) = e*^{−γt}

cos^{2}*(ω*_{j}*t)x*^{2}* _{j}*+ sin

^{2}

*(ω*

_{j}*t)p*

^{2}

*+*

_{j}*sin(2ω*

_{j}*t)*

2 *(x*_{j}*p*_{j}*+ p*_{j}*x** _{j}*)

*+ (N** _{j}*+

^{1}

_{2}

*)(1 − e*

^{−γt}*),*

*p*

^{2}

_{j}*(t) = e*

^{−γt}

sin^{2}*(ω*_{j}*t)x*^{2}* _{j}*+ cos

^{2}

*(ω*

_{j}*t)p*

^{2}

*−*

_{j}*sin(2ω*

*j*

*t)*

2 *(x*_{j}*p*_{j}*+ p*_{j}*x** _{j}*)

*+ (N** _{j}*+

^{1}

_{2}

*)(1 − e*

^{−γt}*),*

*(p*

_{j}*x*

_{j}*(t) + x*

_{j}*p*

_{j}*(t)) = e*

^{−γt}*cos(2ω*

_{j}*t) (x*

_{j}*p*

_{j}*+ p*

_{j}*x*

_{j}*) − sin(2ω*

_{j}*t) x*

^{2}

_{j}*− p*

^{2}

_{j}*.*(57) Note that the evolution keeps locality of the quadratures. On this account, their correlations with the non-local Λ0

*vanishes at any time. In other words for B ∈ {x*^{2}_{j}*, p*^{2}_{j}*, x*_{j}*p*_{j}*+ p**j**x*_{j}*} the response function φ**B**(t) = 0, hence they are*
blind to the perturbation.

On the contrary, for the non-local second moments we have:

*x*_{1}*x*_{2}*(t) = e*^{−γt}

*cos(ω*_{1}*t) cos(ω*_{2}*t)x*_{1}*x*_{2}*+ cos(ω*_{1}*t) sin(ω*_{2}*t)x*_{1}*p*_{2}*+ sin(ω*_{1}*t) cos(ω*_{2}*t)p*_{1}*x*_{2}*+ sin(ω*_{1}*t) sin(ω*_{2}*t)p*_{1}*p*_{2}
*,*
*x*1*p*2*(t) = e*^{−γt}

*− cos(ω*1*t) sin(ω*2*t)x*1*x*2*+ cos(ω*1*t) cos(ω*2*t)x*1*p*2*− sin(ω*1*t) sin(ω*2*t)p*1*x*2*+ sin(ω*1*t) cos(ω*2*t)p*1*p*2

*,*
*p*1*x*2*(t) = e*^{−γt}

*− sin(ω*1*t) cos(ω*2*t)x*1*x*2*− sin(ω*1*t) sin(ω*2*t)x*1*p*2*+ cos(ω*1*t) cos(ω*2*t)p*1*x*2*+ cos(ω*1*t) sin(ω*2*t)p*1*p*2

*,*
*p*1*p*2*(t) = e*^{−γt}

*sin(ω*1*t) sin(ω*2*t)x*1*x*2*− sin(ω*1*t) cos(ω*2*t)x*1*p*2*− cos(ω*1*t) sin(ω*2*t)p*1*x*2*+ cos(ω*1*t) cos(ω*2*t)p*1*p*2

*,*
(58)
*which are all non-local, hence we expect a non-zero response for them. To examine this, let us focus on B = x*1*x*2.

By replacing Eqs. (58) and (54) into the definition of the response function, Eq. (16), we obtain:

*φ**x*_{1}*x*_{2}*(t) = Corr Λ*0*, x*1*x*2*(t)*

0

= e^{−γt}*c*_{1}*(σ*^{∞}_{11})_{J =0}*(σ*_{22}^{∞})_{J =0}*[cos(ω*_{1}*t) cos(ω*_{2}*t) + sin(ω*_{1}*t) sin(ω*_{2}*t)]*

*+ c*2*(σ*_{11}^{∞})_{J =0}*(σ*_{22}^{∞})_{J =0}*[cos(ω*1*t) sin(ω*_{2}*t) − sin(ω*_{1}*t) cos(ω*_{2}*t)]*

=*N*2*− N*1

*γ*^{2}*+ δ*^{2} e^{−γt}*δ cos δt + γ sin δt*

= *N*_{2}*− N*1

p*γ*^{2}*+ δ*^{2}e^{−γt}*cos(δt − θ),* (59)

*where we define θ = arctan γ/δ. By using the same strategy, it is easy to verify that the response function of the*
other non-local observables read as:

*φ*_{x}_{1}_{p}_{2}*(t) =* *N*_{1}*− N*_{2}

*γ*^{2}*+ δ*^{2} e^{−γt}*δ sin δt − γ cos δt =* *N*_{1}*− N*_{2}

p*γ*^{2}*+ δ*^{2}e^{−γt}*sin(δt − θ),*
*φ**p*1*x*2*(t) =* *N*_{2}*− N*1

*γ*^{2}*+ δ*^{2} e^{−γt}*δ sin δt − γ cos δt =* *N*_{2}*− N*1

p*γ*^{2}*+ δ*^{2}e^{−γt}*sin(δt − θ),*
*φ**p*_{1}*p*_{2}*(t) =* *N*2*− N*1

*γ*^{2}*+ δ*^{2} e^{−γt}*δ cos δt + γ sin δt =* *N*2*− N*1

p*γ*^{2}*+ δ*^{2}e^{−γt}*cos(δt − θ).* (60)

### References

[1] H. B. Callen and T. A. Welton,Phys. Rev. 83, 34 (1951).

[2] R. Kubo,Journal of the Physical Society of Japan 12, 570 (1957).

[3] R. Kubo, Reports on Progress in Physics 29, 255 (1966).

[4] D. des Cloizeaux, “Linear response, generalized susceptibility and dispersion theory,” (Interna- tional Atomic Energy Agency, 1968) pp. 325–354.

[5] J. Jensen and A. R. Mackintosh, “Rare earth mag- netism structures and excitations,” (Clarendon Press, 1991).

[6] U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani,Physics Reports 461, 111 (2008).

[7] P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, Nat Phys 12, 778 (2016).

[8] S. Pappalardi, A. Russomanno, A. Silva, and R. Fazio,Journal of Statistical Mechanics: Theory and Experiment 2017, 053104 (2017).

[9] T. Shitara and M. Ueda,Phys. Rev. A 94, 062316 (2016).

[10] W.-L. You, Y.-W. Li, and S.-J. Gu,Phys. Rev. E 76, 022101 (2007).

[11] S.-J. Gu,International Journal of Modern Physics B 24, 4371 (2010).

[12] M. Mehboudi, L. A. Correa, and A. Sanpera,Phys.

Rev. A 94, 042121 (2016).

[13] G. S. Agarwal, Zeitschrift f¨ur Physik A Hadrons and nuclei 252, 25 (1972).

[14] U. Seifert, Reports on Progress in Physics 75, 126001 (2012).

[15] G. Verley, R. Ch´etrite, and D. Lacoste, Journal of Statistical Mechanics: Theory and Experiment 2011, P10025 (2011).

[16] R. Chetrite and S. Gupta, Journal of Statistical Physics 143, 543 (2011).

[17] J. R. Gomez-Solano, A. Petrosyan, S. Ciliberto, R. Chetrite, and K. Gawedzki, Phys. Rev. Lett.

103, 040601 (2009).

[18] R. Chetrite and K. Mallick,Journal of Statistical Physics 148, 480 (2012).

[19] V. V. Albert, B. Bradlyn, M. Fraas, and L. Jiang, Phys. Rev. X 6, 041031 (2016).

[20] M. Baiesi, C. Maes, and B. Wynants,Phys. Rev.

Lett. 103, 010602 (2009).

[21] T. Harada and S.-i. Sasa, Phys. Rev. Lett. 95, 130602 (2005).

[22] J. Prost, J.-F. Joanny, and J. M. R. Parrondo, Phys. Rev. Lett. 103, 090601 (2009).

[23] U. Seifert and T. Speck,EPL (Europhysics Letters) 89, 10007 (2010).

[24] S. L. Braunstein and C. M. Caves,Phys. Rev. Lett.

72, 3439 (1994).

[25] S. L. Braunstein, C. M. Caves, and G. Milburn, Annals of Physics 247, 135 (1996).

[26] M. G. A. Paris,International Journal of Quantum Information 07, 125 (2009).

[27] V. Giovannetti, S. Lloyd, and L. Maccone, Nat Photon 5, 222 (2011).

[28] G. T´oth and I. Apellaniz, Journal of Physics A:

Mathematical and Theoretical 47, 424006 (2014).

[29] V. Giovannetti, S. Lloyd, and L. Maccone,Science 306, 1330 (2004).